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Groupoids and Local Cartesian Closure

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Groupoids and Local Cartesian Closure Groupoids and local cartesian closure Erik Palmgren∗ Department of Mathematics, Uppsala University July 2, 2003 Abstract The 2-category of groupoids, functors and natural isomorphisms is shown to be locally cartesian closed in a weak sense of a pseudo- adjunction. Key words: groupoids, 2-categories, local cartesian closure. Mathematics Subject Classification 2000: 18B40, 18D05, 18A40. 1 Introduction A groupoid is a category in which each morphism is invertible. The notion may be considered as a common generalisation of the notions of group and equivalence relation. A group is a one-object groupoid. Every equivalence relation on a set, viewed as a directed graph, is a groupoid. A recent indepen- dence proof in logic used groupoids. Hofmann and Streicher [8] showed that groupoids arise from a construction in Martin-Löf type theory. This is the so-called identity type construction, which applied to a type gives a groupoid turning the type into a projective object, in the category of types with equiv- alence relations. As a consequence there are plenty of choice objects in this category: every object has a projective cover [12] — a property which seems essential for doing constructive mathematics according to Bishop [1] inter- nally to the category. For some time the groupoids associated with types were believed to be discrete. However, in [8] a model of type theory was given using fibrations over groupoids, showing that this need not be the case. The purpose of this article is to draw some further conclusions for group- oids from the idea of Hofmann and Streicher. Since type theory has a Π- construction, one could expect that some kind of (weak) local cartesian clo- sure should hold for groupoids. The small groupoids can be organised into a category Gpd by taking the morphisms to be functors. This category has many useful properties: it is complete and cocomplete, and unlike the ∗ The author is supported by a grant from the Swedish Research Council (VR). 1 category of groups, it is also cartesian closed [4]. However, it is not lo- cally cartesian closed (see Section 2). The main goal of the present paper is to show that, from a 2-categorical perspective, the groupoids nevertheless satisfy a version of local cartesian closedness. The category Gpd may be re- organised as a 2-category G by letting the natural transformations between functors be the 2-cells, which are then necessarily natural isomorphisms. After some preliminaries on 2-categorical constructions (Section 3 and 4) the main result (Theorem 5.6) is presented. This involves a construction of dependent products of groupoids which is fairly natural, though the verifica- tion of its 2-universal property is quite involved. We should point out that the solution is not completely satisfactory, since the notion of “semi-strict pseudo-adjunction” is probably not general enough. The proofs herein are constructive and choice free, and should be possible to formalise in a topos, or even in predicative versions of toposes. 2 The category of groupoids In this section we see that the category Gpd of small groupoids and functors is not locally cartesian closed. This motivates the laborious 2-categorical constructions in Section 3 and onwards. For a general overview of groupoids we recommend Brown [4]. Higgins [7] contains the most basic and central constructions. The pullback construction will be of special interest, so we state it explicitly here. Let f : A / Z and g : B / Z be functors in Gpd. The pullback of these is constructed analogously to that of sets. Let P be the groupoid where the objects are pairs (x,y), where x is an object of A and y is an object of B such that f(x)= g(y). A morphism from (x,y) to (x′,y′) is a pair of morphisms (ϕ, ψ) where ϕ : x / x′ and ψ : y / y′ are morphisms satisfying f(ϕ)= g(ψ). It follows by the functoriality of f and g that this defines a groupoid. Together with the first and second projection / / functor π1 : P / A and π2 : P / B, this is a pullback of f and g. The coproduct of a set of groupoids can be formed by simply taking the disjoint union of objects and arrows. For the construction of coequalizers, see [7]. It is known that the category Cat of small categories is not locally carte- sian closed (Conduché [5]). Johnstone [9, p. 48] gives a simple argument for this fact, which is easy to adapt to the category Gpd. Let En be the groupoid given by the coarsest equivalence relation on the set Nn = {0, 1,...,n − 1}, that is, for every pair of objects i, j ∈ En, there is a unique arrow eij : i / / j. In the terminology of [7] this is the simplicial groupoid on Nn. 2 1 2 1 h m2 0 f 0 1 m1 1 g 0 0 0 Figure 1 / Let g, h : E2 / E3 be the unique functors such that g(e01)= e01 and / h(e01) = e12. Then the canonical map [g, h] : E2 + E2 / E3 is a regular / epi. To wit, it is the coequalizer of the arrows m1,m2 : E1 / E2 + E2 where m1 maps 0 to 1 of the first summand and m2 maps 0 to 0 of the second / summand. Now let f : E2 / E3 be the unique functor with f(e01)= e02. (See Figure 1.) The pullback of the arrow [g, h] along f is then isomorphic / to k = [k0, k1] : E1 + E1 / E2 where kj is the unique functor mapping the object 0 to j. This k is clearly not epi. Proposition 2.1 The category Gpd is not locally cartesian closed. Proof. Suppose that Gpd is locally cartesian closed. Then since Gpd has coequalizers [7, Ch. 9] it is also regular [9, Lemma 1.5.13]. Consequently, the pullback of a regular epi should be a regular epi. But we have just presented a counter-example to this in Gpd. This is a contradiction. 3 Some 2-categorical notions and constructions Let C be a 2-category. We employ the notation of Borceux [3]. The compo- sition of 1-cells is denoted ◦, and for a 1-cell f : A / B, the left and right / identities are 1A and 1B respectively. A 2-cell α from f : A / B to g : A / B is written α : f ⇒ g. The vertical composition of 2-cells is denoted ⊙. The left and right vertical identities of α are if and ig. The operation ∗ stands for horizontal composition of 2-cells. The left and right horizontal / / identities for α from f : A B to g : A B are i1A and i1B , but we will denote these by iA and iB respectively. Apart from the usual identity and associativity laws these operations satisfy a further identity law and the exchange law ig ∗ if = ig◦f , (γ ⊙ δ) ∗ (α ⊙ β) = (γ ∗ α) ⊙ (δ ∗ β). 3 β α δ γ / for f1 =⇒ f2 =⇒ f3 and g1 =⇒ g2 =⇒ g3, where f1,f2,f3 : A / B, / g1, g2, g3 : B / C. We have distributivity of vertical identities ig ∗ (α ⊙ β) = (ig ∗ α) ⊙ (ig ∗ β) (γ ⊙ δ) ∗ if = (γ ∗ if ) ⊙ (δ ∗ if ), where f : A / B and g : B / C. Inverses can be computed as (α ∗ β)−1 = α−1 ∗ β−1, for invertible α and β. Note that for 2-cells α : h ⇒ k and β : f ⇒ g, where f, g : A / B and h, k : B / C, the interchange law gives the decomposition laws α ∗ β = (α ∗ ig) ⊙ (ih ∗ β), α ∗ β = (ik ∗ β) ⊙ (α ∗ if ). In the following we shall of use the syntactic convention that the operation ∗ binds stronger than ⊙. The 2-category of groupoids to be studied in Sections 4 and 5 is the following. Example 3.1 The 2-category G has small groupoids as objects, functors as 1-cells and natural transformations as 2-cells. The vertical composition ⊙ is the usual vertical composition · of natural transformations. The horizontal composition of α ∗ β (with f,g,h,k as above) is given by (α ∗ β)a = αg(a)h(βa)= k(βa)αf(a). The identity if is the natural transformation if : f ⇒ f defined by (if )a = 1f(a). 3.1 2-slices In an ordinary slice category C/X a morphism from α : A / X to β : B / X is a morphism f : A / B in C satisfying the equality α = β ◦ f. For morphisms between the slices of a 2-category (to be defined below), this equality is replaced by a 2-cell. The specific information given by the 2-cell seems necessary to obtain the pseudo-adjoints of Section 4 and 5. Definition 3.2 Let C be a 2-category. For every object X of C the 2-slice category C//X of C by X is the 2-category given by the following data. (When the intended category is obvious from the context we simply write X¯ for C//X.) 4 • The objects of C//X are arrows (1-cells) of the form α : A / X of C. We also write (α, A) or even α. • The arrows (1-cells) from (α, A) to (β,B) in C//X are pairs f = (f, f) such that f : A / B is an arrow of C and f : α ⇒ β ◦ f is a 2-cell of C. f / A / B ?? ?? ? f ?? =⇒ α ?? β ?? ? X – Suppose that f : (α, A) / (β,B) and g : (β,B) / (γ,C) are 1-cells. Then their composition is given by g ◦ f = (g ◦ f, (g ∗ if ) ⊙ f). (1) – The identity 1(α,A) on (α, A) is (1A, iα).
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